Physics 2011
Chapter 3:
Motion in 2D and 3D
Describing Position in 3-Space
• A vector is used to establish the position of
a particle of interest.
• The position vector, r, locates the particle
at some point in time.
Average Velocity in 3-D
• Vavg = (ř2 – ř1)/(t2-t1) = Δř / Δt
• Δt is scalar so, V vector parallel to ř vector
Instantaneous Velocity
• V’ = lim (Δř / Δt) as Δt  0 = dř / dt
• 3 Components : V’x = dx / dt, etc
• Magnitude, |V’| = SQRT( Vx^2 + Vy ^2 + Vz^2)
Average Acceleration
• âavg = (v’2 – v’1)/(t2-t1) = Δv’ / Δt
• â vector parallel to Δv’ vector
Instantaneous Acceleration
• â = lim (Δv’ / Δt) as Δt  0 = dv’/ dt
• Has the 3 components: âx = d vx/ dt, etc
• These components could also be written
with respect to position vector:
âx = d2x / dt2, etc
Parallel and Perpendicular
Components of Acceleration
Acceleration on Curve
• Different for a) constant speed, b)
increasing speed, c) decreasing speed
Projectile Motion
• Free Fall Problems in 2D or 3D are
“Projectile Motion” problems
• Projectile path is called a Trajectory
Acceleration during Projectile
Motion
• The a vector is constant (g, gravity) and
downward all along the projectile path
2D path, Acceleration Vector
Equations for PM
Uniform Circular Motion
What defines UCM?
• Constant SPEED (not velocity!)
• Constant Radius (R = c)
y
V
(x,y)
R
x
UCM using Polar Coordinates
• The polar coordinate
Cartesian:
Polar:
system (magnitude and
angle) is a natural way of Position: x, y Position: R, θ
describing UCM, where
R and speed are
Velocity:
Velocity:
constant:
Vx = dx/dt,
dR/dt, dθ/dt
Vy = dy/dt
(let ω=dθ/dt)
Vx,Vy always
dR/dt =0
changing
dθ/dt=ω=constant
Velocity in Polar Form:
• Displacement is an Arc, S, of the Circle
• Displacement s = vt (like x = vt + xo)
but s = R = Rt, so:
v = ωR
Average Acceleration in UCM:
• Average Acceleration, aavg = ΔV/Δt
• The Δ V vector points toward origin
Instantanous Acceleration in UCM
• This is called Centripetal Acceleration.
• Like triangles, ΔR and ΔV:
Thus:
v R

v
R
But R = vt for small t
v vt

So:
v
R
a = V2/R
v v 2

t
R
Relative Motion
• First thing: A Frame of Reference
• Since Einstein, a distinction has to be
made between references that behave
classically and those that allow Relativity
• Classical frames of reference are called
Intertial
Inertial Frames of Reference:
• A Reference Frame is the place you measure from.
– It allows you to nail down your (x,y,z) axes
• An Inertial Reference Frame (IRF) is one that is not
accelerating.
– We will consider only IRFs in this course. Stationary or
constant velocity
• Valid IRFs can have fixed velocities with respect to
each other.
– More about this later when we discuss forces.
– For now, just remember that we can make measurements
from different vantage points.
Consider the Frames in Relative
Motion:
• A plane flies due south from Duluth to
MPLS at 100 m/s in a 15 m/s crosswind: